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8/8/2019 Determ.of Eigenvalue & Eigenvector
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Noise and VibrationsNoise and Vibrations(BDC4013)(BDC4013)
DR MUHD HAFEEZ B ZAINULABIDIN
MOHD NO RIHAN B IBRAHIM
Universiti Tun Hussein Onn Malaysia
Determination of Natural Frequencies and Mode ShapesDetermination of Natural Frequencies and Mode Shapes
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2
Necessity to Use Computational MethodNecessity to Use Computational Method
In two degrees of freedom system, solvingthe natural frequencies can be conductedby simply calculating the root of the second
order polynomial.
4 2 0 A B C
By assuming 2 0n n
A B C
Then the natural frequencies can be found
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3
Classical MethodsClassical Methods
Standard Matrix Iteration Method
Dunkerlys Method
Rayleighs Method
Holzers Method
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4
Standard Matrix IterationStandard Matrix Iteration
0 M x K x Considering a generalequation of motion
Assuming harmonicmotion
( ) sin( )i i x t X t
Equation to solve 2 0 M X K X
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Standard Matrix IterationStandard Matrix Iteration(Solution procedures to obtain the lowest(Solution procedures to obtain the lowest natnat freq)freq)
(1) Identify matrix [K] and [M]
(2) Calculate [K] -1
(3) Define the initial trial vector {X} and convergence criteria
(4) Multiply [K] -1 [M] {X} = {Xnew}
(5) Normalized the result {Xnew}/ largest Xnew
(6) Check the convergence , use for a new trial {X}
(7) When it is converged
1n
normalizedX
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Example 1
Find the natural frequenciesand mode shapes of thesystem as shown for
k1=k2=k3=kandm1=m2=m3=mby the matrixiteration method.
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Solution
Flexibility matrix [a]=[k]-1=
Dynamical matrix is
Eigenvalue problem:
321
221
1111
k
321
221
111
1
k
mmk
21and321
221
111
where
mkDXXD
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Solution
1st natural frequency:
Assume Hence
By making the first element equal to unity:
we obtain
1
1
1
1X
6
5
3
12 XDX
m
kX 5773.0,0.3,
0000.2
6667.1
0000.1
0.3 112
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Solution
The various iand are shown below:
The mode shape and natural frequencyconverged in 8 iterations.
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Solution
2nd natural frequency:
Deflated matrix
Let the normalized vector
where must be such that
mXXDD T11112
24698.2
80194.1
00000.11 X
129591.9
24698.2
80194.100000.1
100
010001
24698.2
80194.100000.1
2
211
m
mXmX
T
T
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Solution
=0.32799m-1/2 , hence
73699.0
59102.0
32799.02/11
mX
25768.019921.022048.0
19921.023641.002127.0
22048.002127.045684.0
100
010
001
73699.0
59102.0
32799.0
73699.0
59102.0
32799.0
04892.5
321
221
111
2
T
D
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Solution
Let
By using the iterative scheme, we obtain
1
1
1
1X
25763.0
62885.0
22695.0
00000.1
25763.0
16201.0
05847.0
25763.0
2
2
X
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Solution Continuing the procedure,
Hence 2=0.64307, 2=1.24701
80192.0
44496.0
00000.1
, 2Xm
k
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Solution
3rd natural frequency:
Use a similar procedure as before.
Before computing [D3], need to normalize
59102.0
32794.0
73700.0
giveto 22 XX
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DunkerleysFormula
It gives the approx value of the fundamentalfreq of a composite system.
Consider the following general n DOF system:
For a lumped mass system with diagonalmass matrix, the equation becomes:
01or0 22 maImk
0
0...0
0
0
0...0
...
...
...
10...0
0
10
0...01
1 2
1
21
22221
11211
2
nnnnn
n
n
m
m
m
aaa
aaa
aaa
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DunkerleysFormula
i.e.
Expanding:
0
1...
...1
...1
22211
22222121
12121112
nnnnn
nn
nn
mamama
mamama
mamama
(E.1)0...1
)...
...(
1...
1
2
211,,1212112
11,1313311212211
1
22221112
n
nnnnnn
nnnnnn
n
nnn
n
mmaammaa
mmaammaammaa
mamama
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DunkerleysFormula
Let the roots of this equation be 1/12,1/22,, 1/n2. Thus
Equating coefficients of (1/2)n-1 in (E.1) and(E.2):
In most cases,
(E.2)0...11
...11111
...1111
1
222
2
2
1
2222
2
22
1
2
n
n
n
n
n22211122
2
2
1
m...mm1
...11
nn
n
aaa
n2,3,...,i,11
2
1
2
i
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DunkerleysFormula
Thus
Can also be written as
where in=(1/aiimi)1/2=(kii/mi)1/2
formula)s'(Dunkerley...1 2221112 mamama nni
22
2
2
1
2
1...
111
nnnni
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DunkerlyDunkerly FormulaFormula
(calculation procedures)(calculation procedures)
2 2 2 2
11 22
1 1 1 1
n nn
: fundamental (lowest) natural frequency
n
nn
nn
n
k
m
(1) Identify k11, k22 , knn, m1, m2, mn
(2) Calculate natural frequency of the individual component
(3) Predict the fundamental natural frequency of the system
n nn
: natural frequency of a SDOF system
consisting m and spring of stiffness k
nn
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Example 2
Estimate the fundamental natural frequencyof a simply supported beam carrying 3identical equally spaced masses, as shown
below.
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Problem 1
Estimate the fundamental natural frequenciesof the system shown below. Givenk1=k2=k3=k and m1=m2=m3=m
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Solution
Flexibility matrix
Apply the related equations to solve it.
Does the value of nat. freq. smaller & largerthan the exact valueof nat. frequency?
How many percent ?
321
221111
1
ka
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26
Rayleigh MethodRayleigh Method
This method predicts the fundamental(lowest) natural frequency
This method based on energy method
21
2T mx
212
V kx
1
2
T
T x M x
1
2
T
V x K x
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Rayleigh QuotientRayleigh Quotient
1
2
T
T x M x 1
2
T
V x K x
sin( )
cos( )
x X t
x X t
2max1
2
T
T X M X max1
2
T
V X K X
max maxT V
2
T
T
X K X
X M X
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Rayleigh MethodRayleigh Method
(Calculation procedures)(Calculation procedures)
Identify [K] and [M]
Select any trial vector mode {X}
Predict the fundamental natural frequency
based on the Rayleigh Quotient
2
T
T
X K X
X M X
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Example 3
Estimate the fundamentalfrequency of vibration of thesystem as shown. Assume thatm
1=m
2=m
3=m, k
1=k
2=k
3=k, and
the mode shape is
3
2
1
X
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Solution
Stiffness matrix
Mass matrix
Substitute the assumed mode shape into
110
121012
kk
100
010
001
mm
XR
m
k
m
k
m
k
XR 4629.02143.0
3
2
1
100
010
001
321
3
2
1
110
121
012
321
2
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HolzersMethod
A trial-and-error method to find naturalfrequencies of systems
Requires several trials
The method also gives mode shapes
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HolzerHolzer MethodMethod
1 1 1 1 2
2 2 1 2 1 2 2 3
3 3 3 3 2
( )
( ) ( )
( )
t
t t
t
I k
I k k
I k
2
1 1 1 1 2
2
2 2 1 2 1 2 2 3
2
3 3 3 3 2
( )
( ) ( )
( )
t
t t
t
I k
I k k
I k
2
1
0n
i i
i
I
)cos( tii
Assume
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HolzerHolzer MethodMethod
(calculation)(calculation)
2
1 12 1
1
2
2 3 2 2 1 2 1 2 2
2
1 2 23 2 2 1
2 2
2 2
1 1 2 23 2
2 2
2
3 2 1 1 2 2
2
( )
( )
t
t t t
t
t t
t t
t
I
k
k k k I
k I
k k
I I
k k
I Ik
2
1 1 1 1 2
2
2 2 1 2 1 2 2 3
( )
( ) ( )
t
t t
I k
I k k
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HolzerHolzer MethodMethod
(summary calculation)(summary calculation)
Torsion Translation
1
2
1 12 1
12
3 2 1 1 2 2
2
2 1
1
11
1
2,3,
t
t
i
i i k k
kti
I
k
I Ik
Ik
i n
1
2
1 12 1
12
3 2 1 1 2 2
2
2 1
1
11
1
2,3,
i
i i k k
ki
X
m XX X
k
X X m X m X k
X X m X k
i n
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HolzerHolzer MethodMethod
(calculation procedures)(calculation procedures)
Set initial =0 and set the sweep increment of with a value
Station 1:
X1=1 (or 1=1), calculate M1=2m1X1 (or
2I11)
Station 2:
Calculate X2 (or 2), calculate M2=M1+ 2m2X2 (or
2I22)
Station 3:
Calculate X3 (or 3), calculate M3=M2+ 2m3X3 (or
2I33)
Station n:
Calculate Xn (or n), calculate Mn=Mn-1+ 2mnXn (or
2Inn)
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Example 5
The arrangement of the compressor, turbineand generator in a thermal power plant isshown below. Find the natural frequenciesand mode shapes of the system.
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Solution This is an unrestrained torsional system.
The table below shows its parameters andthe sequence of computations.
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Torsional Systems
The graph below plots the torque Mtappliedat the last disc against the chosen .
The natural frequencies are the at whichMt=0.
The amplitudes i(i=1,2,,n) are the modesha es of the s stem
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Problem 1Problem 1
I1=2 kg m2
I2=4 kg m2
I3=2 kg m2
kt1=3 MNm/ radKt2=2 MNm/ rad
Calculate the naturalfrequencies and modeshapes
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